小尺度发电机Action的磁场放大：依赖于Turbulence模型和雷诺数及普朗特数
Magnetic Field Amplification by Small-Scale Dynamo Action: Dependence on Turbulence Models and Reynolds and Prandtl Numbers
摘要：
小尺度发电机是将湍流动能转化为磁能的过程，因此预计它主要取决于湍流的性质。在这项工作中，我们提出了一个小尺度发电机模型，该模型考虑了湍流速度谱v（l）~ l^θ的斜率，其中l和v（l）是湍流波动的大小和该尺度上的典型速度。磁场的波动分量的时间演化，即，小尺度场由Kazantsev方程描述。我们用量子力学WKB近似求解这个线性微分方程的本征值。这种方法的有效性估计作为一个函数的磁普朗特数Pm。我们计算的最小磁雷诺数发电机行动，Rm_crit，使用我们的湍流速度关联函数模型。对于Kolmogorov湍流（θ =1/3），我们发现临界磁雷诺数约为110，而对于Burgers湍流（θ =1/2），临界磁雷诺数约为2700。在无限磁普朗特数的极限下，我们进一步导出了一般湍流小尺度磁场的增长率为γ ~ Re^（（1-θ）/（1+ θ））.随着磁普朗特数的减小（减小到Pm约大于10），小尺度发电机的增长率减小。这种下降的细节取决于WKB近似，这对于大约为1的磁性普朗特数变得无效。
英文摘要：
The small-scale dynamo is a process by which turbulent kinetic energy is converted into magnetic energy, and thus is expected to depend crucially on the nature of turbulence. In this work, we present a model for the small-scale dynamo that takes into account the slope of the turbulent velocity spectrum v(l) ~ l^theta, where l and v(l) are the size of a turbulent fluctuation and the typical velocity on that scale. The time evolution of the fluctuation component of the magnetic field, i.e., the small-scale field, is described by the Kazantsev equation. We solve this linear differential equation for its eigenvalues with the quantum-mechanical WKB-approximation. The validity of this method is estimated as a function of the magnetic Prandtl number Pm. We calculate the minimal magnetic Reynolds number for dynamo action, Rm_crit, using our model of the turbulent velocity correlation function. For Kolmogorov turbulence (theta=1/3), we find that the critical magnetic Reynolds number is approximately 110 and for Burgers turbulence (theta=1/2) approximately 2700. Furthermore, we derive that the growth rate of the small-scale magnetic field for a general type of turbulence is Gamma ~ Re^((1-theta)/(1+theta)) in the limit of infinite magnetic Prandtl numbers. For decreasing magnetic Prandtl number (down to Pm approximately larger than 10), the growth rate of the small-scale dynamo decreases. The details of this drop depend on the WKB-approximation, which becomes invalid for a magnetic Prandtl number of about unity.
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